Finite Group Databasebeta

← #7625749

Subgroups — #7625749

Order 512.

6836 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3502C211yesmaximal, characteristic
2#34337C218yesmaximal, complemented
3#34336C218yesmaximal, complemented
4#34336C218yesmaximal, complemented
5#34335C218yesmaximal, complemented
6#34271C216yesmaximal
7#34249C212yesmaximal
8#34213C216yesmaximal
9#34182C211yesmaximal, characteristic
10C4×#544C216yesmaximal
11#34074C216yesmaximal
12C4×#519C212yesmaximal
13C2×#2276C211yesmaximal, characteristic
14#3502C4132yescomplemented
15C2×Q82C2214yes—
16C2×#557C2214yes—
17C2×#557C2218yes—
18C2×#544C2216yes—
19C2×#533C2216yes—
20C2×#519C2212yes—
21C2×#519C2212yes—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25#2370C22124yes—
26#2370C22124yes—
27#2370C22124yes—
28#2369C22124yes—
29#2369C22112yes—
30#2365C22112yes—
31#2365C22124yes—
32#2355C22124yes—
33#2355C22112yes—
34#2351C2218yes—
35#2351C2218yes—
36#2351C2218yes—
37Q8×C4⋊C4C2218yes—
38Q8×C4⋊C4C2214yes—
39Q8×C4⋊C4C2218yes—
40Q8×C4⋊C4C2214yes—
41Q8×C4⋊C4C22112yes—
42#2331C22112yes—
43#2331C22112yes—
44#2331C22124yes—
45#2330C22112yes—
46#2330C22112yes—
47#2330C22124yes—
48#2329C2214yes—
49#2329C22112yes—
50#2329C22112yes—
51#2329C22112yes—
52#2317C22112yes—
53#2316C2216yes—
54C4×C4⋊Q8C22124yes—
55C4×#125C22124yes—
56C4×C22⋊Q8C22124yes—
57#2296C22112yes—
58#2295C22124yes—
59#2295C22124yes—
60#2285C2216yes—
61#2276C2218yes—
62#2276C2218yes—
63#2276C2218yes—
64#2276C2218yes—
65#2274C22112yes—
66C42×Q8C2218yes—
67C42×Q8C2218yes—
68C4×#117C22112yes—
69C2×C4×C4⋊C4C2213yes—
70C2×C4×C4⋊C4C2213yes—
71C2×C4×C4⋊C4C2213yes—
72C2×#376C2213yes—
73C23×Q8C2311yescharacteristic
74Q82C2×C4164yescomplemented
75#557C2×C41128yescomplemented
76#557C2×C4164yescomplemented
77#544C2×C4196yes—
78#533C2×C4196yes—
79C2×C4⋊Q8C23124yes—
80C2×C4⋊Q8C2316yes—
81C2×C4⋊Q8C2316yes—
82C2×#125C23124yes—
83C2×#125C23112yes—
84C2×C22⋊Q8C2314yes—
85C2×C22⋊Q8C23112yes—
86#519—216no—
87#519C2×C4132yes—
88C2×C4×Q8—28no—
89C2×C4×Q8C2×C4116yes—
90C2×C4×Q8Q8116yescomplemented
91C2×C4×Q8C2318yes—
92C2×C4×Q8C2318yes—
93C2×C4×Q8C2318yes—
94C2×C4×Q8C23124yes—
95C2×C4×Q8C2314yes—
96C2×C4×Q8C2318yes—
97C2×C4×Q8C2318yes—
98C2×C4×Q8C23112yes—
99C2×C4×Q8C2314yes—
100C2×C4×Q8C23112yes—
101C2×C4×Q8C2318yes—
102C2×#117C23112yes—
103C2×#117C23112yes—
104C22×C4⋊C4C2313yes—
105C22×C4⋊C4C2313yes—
106C22×C4⋊C4C2313yes—
107C22×C42C2313yes—
108C22×C42C2316yes—
109C22×C42C2313yes—
110C22×C42C2313yes—
111#391C23124yes—
112#391C2318yes—
113#391C23124yes—
114#391C23124yes—
115#391C23124yes—
116#389C23124yes—
117#389C23124yes—
118#389C23124yes—
119#389C23124yes—
120#389C23124yes—
121#389C2318yes—
122#389C23124yes—
123#389C23124yes—
124#389C23124yes—
125#389C23112yes—
126#389C23112yes—
127#389C23112yes—
128#389C23112yes—
129#387C23124yes—
130#387C23124yes—
131#387C2318yes—
132#387C23124yes—
133#387C23124yes—
134#387C2318yes—
135#387C23124yes—
136#387C23124yes—
137#387C23148yes—
138#385C23124yes—
139#385C2318yes—
140#384C23124yes—
141#384C2316yes—
142#384C2316yes—
143#382C23124yes—
144#382C23112yes—
145#380C23112yes—
146#380C2314yes—
147C4×C4⋊C4C23112yes—
148C4×C4⋊C4C23124yes—
149C4×C4⋊C4C23124yes—
150C4×C4⋊C4C23148yes—
151C4×C4⋊C4C23112yes—
152C4×C4⋊C4C23112yes—
153C4×C4⋊C4C23112yes—
154C4×C4⋊C4C23112yes—
155C4×C4⋊C4C23124yes—
156C4×C4⋊C4C23124yes—
157C4×C4⋊C4C23124yes—
158C4×C4⋊C4C23112yes—
159C4×C4⋊C4C23112yes—
160C4×C4⋊C4C23112yes—
161C4×C4⋊C4C23124yes—
162C4×C4⋊C4C23124yes—
163C4×C4⋊C4C23124yes—
164C4×C4⋊C4C23124yes—
165C4×C22⋊C4C2318yes—
166C4×C22⋊C4C23124yes—
167#376C23112yes—
168#376C23124yes—
169#376C23112yes—
170C2×#95C2311yescharacteristic
171C2×#95C2313yes—
172C2×#95C2313yes—
173C2×#95C2313yes—
174C43C23124yes—
175C22×Q8C22×C418yes—
176C22×Q8C2×Q818yes—
177C22×Q8#4118yes—
178C22×Q8C24112yes—
179C22×Q8C2414yes—
180C22×Q8C2414yes—
181C23×C4C2413yes—
182C23×C4C2411yescharacteristic
183C23×C4C2413yes—
184C23×C4C2411yescharacteristic
185C23×C4C2413yes—
186C23×C4C2411yescharacteristic
187C23×C4C2413yes—
188C4⋊Q8C22×C4148yes—
189C4⋊Q8C22×C4148yes—
190C4⋊Q8C22×C41192yes—
191#125C22×C4196yes—
192#125C22×C41192yes—
193C22⋊Q8C22×C4132yes—
194C22⋊Q8C22×C4196yes—
195C4×Q8—232no—
196C4×Q8—232no—
197C4×Q8—232no—
198C4×Q8—824no—
199C4×Q8C22×C4132yes—
200C4×Q8C22×C4132yes—
201C4×Q8C22×C4164yes—
202C4×Q8C22×C4164yes—
203C4×Q8C22×C4196yes—
204C4×Q8C22×C4196yes—
205C4×Q8C2×Q8164yes—
206#117—248no—
207#117C22×C4196yes—
208C2×C4⋊C4—212no—
209C2×C4⋊C4C22×C4124yes—
210C2×C4⋊C4C2×Q8124yes—
211C2×C4⋊C4C24112yes—
212C2×C4⋊C4C24112yes—
213C2×C4⋊C4C24112yes—
214C2×C4⋊C4C24112yes—
215C2×C4⋊C4C24124yes—
216C2×C4⋊C4C2414yes—
217C2×C4⋊C4C2414yes—
218C2×C4⋊C4C24112yes—
219C2×C4⋊C4C24124yes—
220C2×C4⋊C4C2418yes—
221C2×C4⋊C4C2418yes—
222C2×C4⋊C4C24112yes—
223C2×C4⋊C4C24112yes—
224C2×C4⋊C4C24112yes—
225C2×C4⋊C4C2418yes—
226C2×C4⋊C4C2414yes—
227C2×C4⋊C4C24112yes—
228C2×C4⋊C4C24124yes—
229C2×C4⋊C4C24112yes—
230C2×C22⋊C4C2418yes—
231C2×C22⋊C4C2418yes—
232C2×C42—212no—
233C2×C42—86no—
234C2×C42C22×C4124yes—
235C2×C42C2×Q8124yes—
236C2×C42C24124yes—
237C2×C42C24112yes—
238C2×C42C24124yes—
239C2×C42C24124yes—
240C2×C42C2418yes—
241C2×C42C2418yes—
242C2×C42C24124yes—
243C2×C42C24124yes—
244C2×C42C2416yes—
245C2×C42C2416yes—
246C2×C42C24112yes—
247#95C2412yes—
248#95C2412yes—
249#95C2416yes—
250#95C24112yes—
251#95C24112yes—
252#95C2416yes—
253#95C2418yes—
254#95C2418yes—
255#95C24124yes—
256#95C24124yes—
257#95C2418yes—
258#95C24124yes—
259#95C24124yes—
260#95C24124yes—
261#95C2418yes—
262#95C2418yes—
263C24C2511yescharacteristic
264C2×Q8—412no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—412no—
268C2×Q8C4×Q8164yescomplemented
269C2×Q8C23×C4116yes—
270C2×Q8C23×C4148yes—
271C2×Q8C23×C4116yes—
272C2×Q8C22×Q8116yes—
273C2×Q8C2×#41116yes—
274C22×C4—22no—
275C22×C4—26no—
276C22×C4—43no—
277C22×C4—43no—
278C22×C4—43no—
279C22×C4—43no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—43no—
285C22×C4—43no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4C23×C414yes—
289C22×C4C23×C4112yes—
290C22×C4C23×C4112yes—
291C22×C4C22×Q814yes—
292C22×C4C22×Q8112yes—
293C22×C4C22×Q8112yes—
294C22×C4C2×#41112yes—
295C22×C4C2512yes—
296C22×C4C2518yes—
297C22×C4C2516yes—
298C22×C4C2518yes—
299C22×C4C2511yescharacteristic
300C22×C4C2518yes—
301C22×C4C2518yes—
302C22×C4C2511yescharacteristic
303C22×C4C2516yes—
304C22×C4C2512yes—
305C22×C4C2516yes—
306C22×C4C2516yes—
307C4⋊C4—224no—
308C4⋊C4—224no—
309C4⋊C4—224no—
310C4⋊C4—48no—
311C4⋊C4—424no—
312C4⋊C4—48no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—412no—
316C4⋊C4—412no—
317C4⋊C4—424no—
318C4⋊C4C23×C4148yes—
319C4⋊C4C23×C4148yes—
320C4⋊C4C23×C4148yes—
321C4⋊C4C23×C4132yes—
322C4⋊C4C23×C4116yes—
323C4⋊C4C23×C4148yes—
324C4⋊C4C23×C4196yes—
325C4⋊C4C22×Q8148yes—
326C22⋊C4—216no—
327C22⋊C4C23×C4132yes—
328C42—224no—
329C42—248no—
330C42—424no—
331C42—424no—
332C42—48no—
333C42—48no—
334C42—424no—
335C42C23×C4124yes—
336C42C23×C4124yes—
337C42C23×C4196yes—
338C42C22×Q8148yes—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23C24×C412yes—
344C23C23×Q812yes—
345C23C22×#4112yes—
346C23C2611yescharacteristic
347Q8—416no—
348Q8—448no—
349Q8C2×C4×Q8164yes—
350C2×C4—26no—
351C2×C4—26no—
352C2×C4—28no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—26no—
356C2×C4—28no—
357C2×C4—28no—
358C2×C4—28no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—26no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—28no—
366C2×C4—28no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—28no—
370C2×C4—28no—
371C2×C4—28no—
372C2×C4—28no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—26no—
376C2×C4—44no—
377C2×C4—412no—
378C2×C4—44no—
379C2×C4—412no—
380C2×C4—44no—
381C2×C4—412no—
382C2×C4C2×C4×Q8148yes—
383C2×C4C24×C412yes—
384C2×C4C24×C412yes—
385C2×C4C24×C4116yes—
386C2×C4C24×C4116yes—
387C2×C4C24×C4112yes—
388C2×C4C24×C4112yes—
389C2×C4C23×Q814yes—
390C2×C4C23×Q8112yes—
391C2×C4C23×Q8112yes—
392C2×C4C22×#41112yes—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22C22×C4×Q814yes—
400C22C25×C411yescharacteristic
401C22C24×Q811yescharacteristic
402C22C23×#4111yescharacteristic
403C22H(7,𝔽2)11yescharacteristic
404C222−1+611yescharacteristic
405C222−1+611yescharacteristic
406C222−1+611yescharacteristic
407C4—24no—
408C4—212no—
409C4—22no—
410C4—22no—
411C4—216no—
412C4—212no—
413C4—216no—
414C4—48no—
415C4—42no—
416C4—48no—
417C4—46no—
418C4C22×C4×Q8124yes—
419C2—24no—
420C2C23×C4×Q811yescharacteristic
421C2#6265311yescharacteristic
422C2#6265411yescharacteristic
423C2#6284911yescharacteristic
424C2#6285011yescharacteristic
425C2#6285111yescharacteristic
426C2#6285411yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC23C26
commutatorC22C25×C4
prime radicalC24C25
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #34337C2semidirect
3. #34336C2semidirect
4. #34336C2semidirect
5. #34335C2semidirect
14. #3502C4semidirect
74. Q82C2×C4semidirect
75. #557C2×C4semidirect
76. #557C2×C4semidirect
90. C2×C4×Q8Q8semidirect
268. C2×Q8C4×Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#35022
#343372
#343362
#343352
#342712
#342492
#342132
#341822
C4×#5442
#340742
C4×#5192
C2×#22762